[FOM] Why not NF?

[Thomas Forster]

On Sun, 19 Feb 2006, Martin Davis wrote:

> Hendrik wrote:
> 
>  >Martin Davis wrote:
>  >> NF suffers from at least two grave faults:
>  > >1. It's inconsistent with AC
>  > >2. Cantor's theorem 2^x > x fails.
> 
>  >And why are those two sacred?
> 
> Sacred? Not at all. An integral part of contemporary mathematics? Yes.

   "AC is an integral part of contemporary mathematics".  Well, yes, but 
then so is ~AC.  Unless of course you want to dismiss all of constructive 
mathematics.   I think the point could be better made by saying that AC 
fails in NF for reasons that don't seem to be mathematically substantial 
or interesting - unlike the failure of AC in constructive maths.  Nobody
really understand what is going on there, and i don't know anybody who 
even thinks they understand it.

   The reason why i study NF (apart from habit!) is that it is a very 
interesting system, with lots of mysteries - rather than that it is a 
suitable basis for mathematics.  If i were to argue for NF as 
a basis for mathematics i could do so on the grounds that there is no 
reason to suppose that the theory of wellfounded-sets-in-NF contradicts 
ZFC.  In other words, if NF is consistent, you get not only ZF but 50% 
Extra - free! a rudimentary theory of Big sets (V, On, etc).  How good an 
idea this is is not clear yet.  There are examples known of sets of 
naturals definable with these big sets as parameters, and these of course 
have no correlates in ZF.  This will probably turn out to be no more than 
a curiosity, but one never knows, and nobody has investiaged it so far.

      tf



  URL: www.dpmms.cam.ac.uk/~tf   Tel: +44-1223-337981
  (U Cambridge); +44-20-7882-3659 (QMW) +44-7887-701-562
 (mobile)